The basic structure and basic mode of operation of a stabilizer for a motor vehicle are known from practice. Thus, a stabilizer serves to improve the road holding. A stabilizer comprises a torsion bar with longitudinal links connected on both sides of the torsion bar, the torsion bar itself being connected with a body of a motor vehicle and the longitudinal links being connected with the wheel suspensions. The torsion bar of the stabilizer is also known as the stabilizer tube and the longitudinal links of the stabilizer are also known as anti-roll bars. From practice it is already known to weld the longitudinal links or anti-roll bars to the torsion bar or stabilizer tube. For this, in stabilizers known from practice the procedure adopted is first to weld the longitudinal links made from a spring steel to the torsion bar also made from a spring steel, and then heat treat the welded unit consisting of the longitudinal links and the torsion bar in a hardening furnace. When after the welding of the longitudinal links to the torsion bar the stabilizer is heat treated as a unit, relatively large hardening furnaces are needed. This is disadvantageous.